Improved Measurement of the CKM Angle alpha Using B0->rho+rho- Decays

We present results from an analysis of B0 ->rho+rho- using 232 million Upsilon(4S) decays collected with the BABAR detector at the PEP-II asymmetric-energy $B$ Factory at SLAC. We measure the longitudinal polarization fraction f_L = 0.978 +- 0.014 (stat) +0.021 -0.029 (syst) and the CP-violating parameters SLong = -0.33 +- 0.24 (stat) +0.08 -0.14 (syst) and CLong = -0.03 +- 0.18 (stat) +- 0.09 (syst). Using an isospin analysis of B ->rho rho decays we determine the unitarity triangle alpha. The solution compatible with the Standard Model is alpha = (100 +- 13) degrees.

PACS numbers: 13.25.Hw,12.15.Hh,11.30.Er In the Standard Model, CP -violating effects in the Bmeson system arise from a single phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix [1]. Interference between direct decay and decay after B 0 B 0 mixing in B 0 (B 0 ) → ρ + ρ − results in a time-dependent decay-rate asymmetry that is sensitive to the angle α ≡ arg [−V td V * tb /V ud V * ub ] in the unitarity triangle of the CKM matrix . This decay proceeds mainly through a b → uud tree diagram. The presence of penguin loop contributions introduces additional phases that shift the experimentally measurable parameter α eff away from the value of α. However, measurements of the B + → ρ + ρ 0 branching fraction and the upper limit for B 0 → ρ 0 ρ 0 [2,3] show that the penguin contribution in B → ρρ is small with respect to the leading tree diagram, and δα ρρ = α eff −α is constrained at ±11 • at 1σ [3]. This Letter presents an update of the time-dependent analysis of B 0 (B 0 ) → ρ + ρ − and measurement of the CKM angle α reported in [4].
The CP analysis of B decays to ρ + ρ − is complicated by the presence of a mode with longitudinal polarization and two with transverse polarizations. The longitudinal mode is CP even, while the transverse modes contain CP -even and CP -odd states. Empirically, the decay is observed to be dominated by the longitudinal polarization [4], with a fraction f L defined by the fraction of the helicity zero state in the decay. The angular distribution is where θ i=1,2 is the angle between the π 0 momentum and the direction opposite the B 0 in the ρ rest frame, and we have integrated over the angle between the ρ decay planes. The analysis reported here is improved over our earlier publication [4] by a change in selection requirements resulting in an increased signal efficiency; introduction of a signal time dependence that accounts for possible misreconstruction; and use of a more detailed background model. This measurement uses 232 million Υ (4S) → BB decays collected with the BABAR [5] detector at the PEP-II asymmetric-energy B Factory at SLAC.
We reconstruct B 0 (B 0 ) → ρ + ρ − candidates (B rec ) from combinations of two charged tracks and two π 0 candidates. We require that both tracks have particle identification information inconsistent with the electron, kaon, and proton hypotheses. The π 0 candidates are formed from pairs of photons each of which has a measured energy greater than 50 MeV. The reconstructed π 0 mass must satisfy 0.10 < m γγ < 0.16 GeV/c 2 . The mass of the ρ candidates must satisfy 0.5 < m π ± π 0 < 1.0 GeV/c 2 . When multiple B candidates can be formed, we select the one that minimizes the sum of (m γγ − m π 0 ) 2 where m π 0 is the true π 0 mass. If more than one candidate has the same π 0 mesons, we select one at random.
Combinatorial backgrounds dominate near | cos θ i | = 1, and backgrounds from B decays tend to concentrate at negative values of cos θ i . We reduce these backgrounds with the requirement −0.90 < cos θ i < 0.98.
Continuum e + e − → qq (q = u, d, s, c) events are the dominant background. This background is reduced by requiring that | cos B T R | < 0.8, where B T R is the angle between the B thrust axis and that of the rest of the event, ROE. The thrust axis of the B is the direction which maximizes the longitudinal momenta of the particles in the B candidate. To distinguish signal from continuum we use a neural network (N ) to combine ten discriminating variables: the event shape variables that are used in the Fisher discriminant in Ref [6]; the cosine of the angle between the direction of the B and the collision axis (z) in the e + e − center-of-mass (CM) frame; the cosine of the angle between the B thrust axis and the z axis, | cos B T R |; the decay angle of each π 0 (defined in analogy to the ρ decay angle, θ i ); and the sum of transverse momenta in the ROE relative to the z axis.
Signal events are identified kinematically using two variables, the difference ∆E between the CM energy of the B candidate and √ s/2, and the beam-energysubstituted mass √ s is the total CM energy. The B momentum p B and four-momentum of the initial state (E i , p i ) are defined in the laboratory frame. We accept candidates that satisfy 5.23 < m ES < 5.29 GeV/c 2 and −0.12 < ∆E < 0.15 GeV. The asymmetric ∆E selection reduces background from higher-multiplicity B decays.
To study the time-dependent asymmetry one needs to measure the proper-time difference, ∆t, between the two B decays in the event, and to determine the flavor of the other B meson (B tag ). We calculate ∆t from the measured separation ∆z between the B rec and B tag decay vertices [7]. We determine the B rec vertex from the two charged-pion tracks in its decay. The B tag decay vertex is obtained by fitting the other tracks in the event, with constraints from the B rec momentum and the beam-spot location. The RMS resolution on ∆t is 1.1 ps. We only use events that satisfy |∆t| < 20 ps and for which the error on ∆t less than 2.5 ps. The flavor of the B tag meson is determined with a multivariate technique [6] that has a total effective tagging efficiency of (29.9 ± 0.5)%.
Signal candidates may pass the selection requirement even if one or more of the pions assigned to the ρ + ρ − state belongs to the other B in the event. These selfcross-feed (SCF) candidates constitute 50% (26%) of the accepted signal for f L = 1 (f L = 0). The majority of SCF events have both charged pions from the ρ + ρ − final state, and unbiased CP information (correct-track SCF). There is a SCF component (14% of the signal) where at least one track in B rec is from the rest of the event. These wrong track events have biased CP information, and are treated separately for the CP result. The probability density function (PDF) describing wrong track events is used only in determining the signal yield and polarization. A systematic error is assigned to the CP results from this type of signal event.
Each candidate is described with the eight B rec kinematic variables: m ES , ∆E, the m π ± π 0 and cos θ i values of the two ρ mesons, ∆t, and N . For each fit component, we construct a PDF that is the product of PDFs for these variables, neglecting correlations. This introduces a fit bias that is corrected with the use of Monte Carlo (MC) simulation. The continuum-background yield and its PDF parameters for m ES , ∆E, cos θ i , and N are floated in the fit to data. The continuum m π ± π 0 distribution is described by a Breit-Wigner and polynomial shape, and is derived from m ES and ∆E data sidebands. For all other fit components the PDFs are extracted from high-statistics MC samples. The cos θ i distributions for the background are described by a non-parametric (NP) PDF derived from the MC samples, as the detector acceptance and selection modify the known vector-meson decay distribution. The true signal distribution is given by Eq. 1 multiplied by an acceptance function determined from signal MC samples, whereas SCF signal is modeled using NP PDFs.
The signal decay-rate distribution for both polariza- where τ is the mean B 0 lifetime, ∆m d is the B 0 B 0 mixing frequency, and S = S L or S T and C = C L or C T are the CP -asymmetry parameters for the longitudinally and transversely polarized signal. The parameters S and C describe B-mixing induced and direct CP violation, respectively. S and C for the longitudinally polarized wrong-track signal are fixed to zero. The ∆t PDF takes into account incorrect tags and is convolved with the resolution function described below. Since f L is approximately 1, the fit has no sensitivity to either S T or C T . We set these parameters to zero and vary them in the evaluation of systematic uncertainties. The signal ∆t resolution function consists of three Gaussians (∼90% core, ∼9% tail, ∼1% outliers), and takes into account the per-event error on ∆t from the vertex fit. The resolution is parameterized using a large sample of fully reconstructed hadronic B decays [7]. For wrong-track SCF we replace the B-meson lifetime by an effective lifetime obtained from MC simulation to account for the difference in the resolution. The nominal ∆t distribution for the B backgrounds is a NP representation of the MC samples; in the study of systematic errors we replace this model with the one used for signal. The resolution for continuum background is described by the sum of three Gaussian distributions whose parameters are determined from data.
We perform an unbinned extended maximum likelihood fit. The results of the fit are 617 ± 52 signal events, after correction of a 68 event fit bias, with f L = 0.978 ± 0.014, S L = −0.33 ± 0.24 and C L = −0.03 ± 0.18. The measured signal yield, polarization, and CP parameters are in agreement with our earlier publication [4], with significantly improved precision. Figure 1 shows distributions of m ES , ∆E, cos θ i and m π ± π 0 for the highest purity tagged events with a loose requirement on N . The plot of m ES contains 14% of the signal and 1.5% of the background. For the other plots there is an added constraint that m ES > 5.27 GeV/c 2 ; these requirements retain 11.5% of the signal and 0.4% of the background. Figure 2 shows the ∆t distribution for B 0 and B 0 tagged events. The time-dependent decay-rate asymmetry [N (∆t) − N (∆t)]/[N (∆t) + N (∆t)] is also shown, where N (N ) is the decay-rate for B 0 (B 0 ) tagged events.
We have studied possible sources of systematic uncertainties on f L , S L and C L . The dominant uncertain-ties for f L come from floating the B background yields ( +0.00 −0.02 ), non-resonant events (0.015) and fit bias (0.01).
The dominant systematic uncertainty on the CP results comes from the uncertainty in the B-background branching ratios. This results in a shift on S L (C L ), as large as +0.00 −0.12 ( +0.008 −0.003 ). Additional uncertainties on the CP results come from possible CP violation in the B background, calculated as in Ref. [4]. We allow for a CP asymmetry up to 20% in B decays to final states with charm, resulting in an uncertainty of 0.027 (0.045) on S L (C L ). Allowing for possible CP violation in the transverse polarization results in an uncertainty of 0.02 ( +0.002 −0.016 ) on S L (C L ). We estimate the systematic error on our CP results from neglecting the interference between B 0 (B 0 ) → ρ + ρ − and other 4π final states: B → a 1 π, ρππ 0 and B → πππ 0 π 0 . Strong phases and CP content of the interfering states are varied between zero and maximum using uniform prior distributions, and the RMS deviation of the parameters from nominal is taken as the systematic error; this is found to be 0.02 on S L and C L . Other contributions that are large include knowledge of the vertex detector alignment 0.034 (0.005) on S L (C L ), and possible CP violation in the doubly-Cabibbo-suppressed decays on the tag side of the event [10]. We allow CP violation in the wrong-track SCF to vary between −1 and +1, which results in changes of 0.007 (0.012) in S L (C L ). The nominal fit does not account for non-resonant background. If we add a non-resonant component of B → ρππ 0 events to the likelihood, we fit 83 ± 59 non-resonant events and observe only a (6 ± 4)% drop in signal yield. This effect is included in our total systematic uncertainty. Possible contributions from σ(400)π 0 π 0 decays are neglected due to the small reconstruction efficiency (0.4%). Our results are f L = 0.978 ± 0.014(stat) +0.021 −0.029 (syst), S L = −0.33 ± 0.24(stat) +0.08 −0.14 (syst), C L = −0.03 ± 0.18(stat) ± 0.09(syst), where the correlation between S L and C L is −0.042.
We constrain the CKM angle α from an isospin analysis [11] of B → ρρ. The inputs to the isospin analysis are the amplitudes of the CP -even longitudinal polarization of the ρρ final state, as well as the measured values of S L and C L for B 0 (B 0 ) → ρ + ρ − . We use the measurements of f L , S L and C L presented here; the branching fraction of B 0 → ρ + ρ − from [4], which uses information from [12]; the combined branching fraction and f L for B → ρ + ρ 0 from Ref. [2]; the central value corresponding to the upper limit of B(B → ρ 0 ρ 0 ) from Ref. [3]. We ignore electroweak penguins and possible I = 1 amplitudes [13].
To interpret our results in terms of a constraint on α from the isospin relations, we construct a χ 2 that includes the measured quantities expressed as the lengths of the sides of the isospin triangles and we determine the minimum χ 2 0 . As the isospin triangles do not close with the current central values of the branching ratios, we have adopted a toy MC techniques to compute the confidence level (CL) on α; our method is similar to the approach proposed in Ref. [14]. For each value of α, scanned between 0 and 180 • , we determine the difference ∆χ 2 DATA (α) between the minimum of χ 2 (α) and χ 2 0 . We then generate MC experiments around the central values obtained from the fit to data with the given value of α and we apply the same procedure. The fraction of these experiments in which ∆χ 2 MC (α) is smaller than ∆χ 2 DATA (α) is interpreted as the CL on α. Figure 3 shows 1 − CL for α obtained from this method. Selecting the solution closest to the CKM combined fit average [15,16] we find α = 100 • ± 13 • , where the error is dominated by δα ρρ which is ±11 • at 1σ. The 90% CL allowed interval for α is between 79 • and 123 • . : CL on α obtained from the isospin analysis with the statistical method described in [15]. The dashed lines correspond to the 68% (top) and 90% (bottom) CL intervals.
In summary we have improved the measurement of the CP -violating parameters S L and C L in B 0 (B 0 ) → ρ + ρ − using a data-sample 2.6 times larger than that in Ref. [4]. We do not observe mixing-induced or direct CP violation. We derive a model-independent measurement of the CKM angle α, which is the most precise to date.
We are grateful for the excellent luminosity and machine conditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and